Related papers: Wandering Fatou components on p-adic polynomial dy…

We first establish any continuum without interiors can be a limit set of iterations of an entire function on an oscillating wandering domain, and hence arise as a component of Julia sets. Recently, Luka Boc Thaler showed that every bounded…

We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it…

We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in…

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings $D$. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda)=\lambda$ for any $n \in \mathbb{N}$, where…

We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $\lambda$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference…

Let D be the open unit disc in C. The paper deals with the following conjecture: If f is a continuous function on bD such that the change of argument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1 has no zero on bD…

We study the dynamics of a generic endomorphism $f$ of an Oka-Stein manifold $X$. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several…

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials $\{P_n\}$ to properties of the support. More precisely we relate the Julia…

We discuss the dynamics of semigroups of transcendental entire functions using Fatou-Julia theory and provide a condition for the complete invariance of escaping set and Julia set of transcendental semigroups. Results regarding limit…

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$…

It has been known since Julia that polynomials commuting under composition have the same Julia set. More recently in the works of Baker and Eremenko, Fern\'andez, and Beardon, results were given on the converse question: When do two…

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding.…

Let f be a polynomial endomorphism of degree d>1 of C^k (k>1) which extends to a holomorphic endomorphism of P^k. Assume that the maximal order Julia set of f is laminated by real hypersurfaces in some open set. We show that f is homogenous…

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ <p, r>_S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr…

In this work, orthogonal polynomials satisfying $R_I$ type recurrence relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-\lambda_n (z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and $\mathcal{P}_0(z) = 1$ are analyzed…

In this paper we discuss the boundedness of the Fatou components for the sine family and the extended sine family, mainly when the parameter \lambda has modulus greater than 1 and the map is post-critically bounded.

We estimate the number $|\mathcal{A}_{\boldsymbol\lambda}|$ of elements on a nonlinear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $r$ having factorization pattern…

Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of…

We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove…

We consider a mechanism for area preserving Hamiltonian systems which leads to the enhanced probability, $P(\lambda, t)$, to find small values of the finite time Lyapunov exponent, $\lambda$. In our investigation of chaotic dynamical…